Robustness and Directed Structures in Ecological Flow Networks

Robustness of ecological flow networks under random failure of arcs is considered with respect to two different functionalities: coherence and circulation. In our previous work, we showed that each functionality is associated with a natural path notion: lateral path for the former and directed path for the latter. Robustness of a network is measured in terms of the size of the giant laterally connected arc component and that of the giant strongly connected arc component, respectively. We study how realistic structures of ecological flow networks affect the robustness with respect to each functionality. To quantify the impact of realistic network structures, two null models are considered for a given real ecological flow network: one is random networks with the same degree distribution and the other is those with the same average degree. Robustness of the null models is calculated by theoretically solving the size of giant components for the configuration model. We show that realistic network structures have positive effect on robustness for coherence, whereas they have negative effect on robustness for circulation. Introduction Networks have been usually considered as undirected in the field of complex networks (Newman, 2003). However, many real-world networks are directed so that the direction of interaction is important for the functioning of the systems. Recently, it has been revealed that directed networks have richer structures such as directed assortativity (Foster et al., 2010) and flow hierarchy (Mones, 2013). In our previous work, we proposed a new path notion involving directedness called lateral path that can be seen as the dual notion to the usual directed path (Haruna, 2011). Based on category theoretic formulation, we derived the lateral path as a natural path notion associated with the dynamic mode of biological networks: a network is a pattern constructed by gluing functions of entities constituting the network (Haruna, 2012). Thus, its functionality is coherence, whereas the functionality of the directed path is transport. We showed that there is a division of labor with respect to the two functionalities within a network for several types of biological networks: gene regulation, neuronal and ecological ones (Haruna, 2012). It was suggested that the two complementary functionalities are realized in biological systems by making use of the two ways of tracing on a directed network, namely, lateral and directed. In this paper, we address robustness of ecological flow networks with respect to the lateral path and directed path, respectively. Since the natural connectedness notion associated with the directed path is the strong connectedness, we consider robustness of the giant strongly connected component (GSCC) for the latter. For the former, robustness of the giant lateral connected component (GLCC) is of interest. Thus, we assess robustness of ecological flow networks in terms of two different functionalities, namely, coherence and circulation, both of which are important for the functioning of them (Ulanowicz, 1997). Robustness of ecological networks is an intriguing issue in recent studies (Montoya et al., 2006; Bascompte, 2009). Initially, robustness of general complex networks has been argued qualitatively in terms of critical thresholds for the existence of the giant component (Albert et al., 2000; Cohen et al., 2001). For ecological networks, their robustness has been measured by the size of secondary extinctions (Solé and Montoya, 2001; Dunne et al., 2002). Here, we employ a recently proposed idea to measure robustness quantitatively (Schneider et al., 2011; Herrmann et al., 2011). As a first step, we consider only random failure of arcs. The size of giant components is measured by the number of arcs involved because laterally connected components are defined only on the set of arcs. Here, we study the impact of realistic network structures on robustness with respect to the two functionalities. Two complementary measures of it are proposed by comparing the robustness of a given real network with that of the two null models: random networks with the same degree distribution and those with the same average degree. The robustness of the two null models is calculated by theoretically solving the percolation problem on the configuration model, random networks with an arbitrary degree distribution (Newman et al., 2001). This paper is organized as follows. In Section 2, we develop a theory to calculate the size of GLCC and GSCC unFigure 1: An example of lateral path. der random removal of arcs in the configuration model. In Section 3, we propose two measure for the impact of realistic structures on robustness of networks by using the theoretical result obtained in Section 2. In Section 4, the proposed measures are applied to 10 ecological flow networks. In Section 5, we discuss the results and indicate future directions. Random Removal of Arcs in the Configuration Model In this section, we consider a percolation problem, random removal of arcs, in the configuration model with respect to the lateral connectedness and the strong connectedness. A lateral path in a directed network is a path in the network such that the direction of arcs involved changes alternately (Haruna, 2012) (Fig. 1). Two arcs are called laterally connected if they are connected by a lateral path (Haruna, 2011). Lateral connectedness defines an equivalence relation on the set of arcs. Each equivalence class is called laterally connected component. Since lateral connectedness is defined on the set of arcs, here we also consider strong connectedness for arcs. Two arcs are called strongly connected if there is a directed path from one arc to the other arc, and vice versa. Let us consider a random directed network with degree distribution P (ki, ko). P (ki, ko) is the fraction of nodes in the network with in-degree ki and out-degree ko. We make use of the generating function formalism (Callaway et al., 2000; Newman et al., 2001) to calculate the sizes of giant laterally or strongly connected components (in short, GLCC or GSCC, respectively) after removing arcs uniformly at random with probability 1−φ, where φ is the occupation probability. The generating function for P (ki, ko) is